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  1. ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡ ♦♥ ❛❝❝♦r❞✐♦♥ ❝♦♠♣❧❡①❡s ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ✼✽ è♠❡ ❙é♠✐♥❛✐r❡ ❧♦t❤❛r✐♥❣✐❡♥ ❞❡ ❝♦♠❜✐♥❛t♦✐r❡ ▼❛r❝❤ ✷✾ t❤ ✱ ✷✵✶✼

  2. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❉✐ss❡❝t✐♦♥ ❂ s❡t ♦❢ ♣❛✐r✇✐s❡ ♥♦♥❝r♦ss✐♥❣ ❞✐❛❣♦♥❛❧s ❚r✐❛♥❣✉❧❛t✐♦♥ ❂ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❞✐ss❡❝t✐♦♥ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  3. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❈❡❧❧ ❂ ❜♦✉♥❞❡❞ ❝♦♥♥✳ ❝♦♠♣✳ ♦❢ t❤❡ ❝♦♠♣❧❡♠❡♥t ❚r✐❛♥❣✉❧❛t✐♦♥ ❂ ❛❧❧ ❝❡❧❧s ❛r❡ tr✐❛♥❣❧❡s ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  4. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❈♦♥s✐❞❡r ✐♥t❡r❧❛❝❡❞ r❡❞ ❛♥❞ ❜❧✉❡ ♣♦❧②❣♦♥s 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  5. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❋✐① ❛ r❡❢❡r❡♥❝❡ r❡❞ ❞✐ss❡❝t✐♦♥ ❉ ◦ 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  6. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❉ ◦ ✲❛❝❝♦r❞✐♦♥ ❞✐❛❣♦♥❛❧ = ❜❧✉❡ ❞✐❛❣♦♥❛❧ ❝r♦ss✐♥❣ ❛ ✏❜❧✉❡ ❞✐❛❣♦♥❛❧✑ ❝♦♥♥❡❝t❡❞ s❡t ♦❢ r❡❞ ❞✐❛❣♦♥❛❧s 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  7. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❉ ◦ ✲❛❝❝♦r❞✐♦♥ ❞✐❛❣♦♥❛❧ = ❜❧✉❡ ❞✐❛❣♦♥❛❧ ❝r♦ss✐♥❣ ❛ ✏❜❧✉❡ ❞✐❛❣♦♥❛❧✑ ❝♦♥♥❡❝t❡❞ s❡t ♦❢ r❡❞ ❞✐❛❣♦♥❛❧s 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ NO! 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  8. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ▼❛①✐♠❛❧ ❉ ◦ ✲❛❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥ = ✐♥❝❧✉s✐♦♥ ♠❛①✳ ❞✐ss❡❝t✐♦♥ ✏❜❧✉❡ ❞✐ss❡❝t✐♦♥✑ ❝♦♥t❛✐♥✐♥❣ ❜❧✉❡ ❞✐❛❣♦♥❛❧s 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  9. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ▼❛①✐♠❛❧ ❉ ◦ ✲❛❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥ = ✐♥❝❧✉s✐♦♥ ♠❛①✳ ❞✐ss❡❝t✐♦♥ ✏❜❧✉❡ ❞✐ss❡❝t✐♦♥✑ ❝♦♥t❛✐♥✐♥❣ ❜❧✉❡ ❞✐❛❣♦♥❛❧s 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  10. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ▼❛①✐♠❛❧ ❉ ◦ ✲❛❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥ = ✐♥❝❧✉s✐♦♥ ♠❛①✳ ❞✐ss❡❝t✐♦♥ ✏❜❧✉❡ ❞✐ss❡❝t✐♦♥✑ ❝♦♥t❛✐♥✐♥❣ ❜❧✉❡ ❞✐❛❣♦♥❛❧s 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  11. ❜❧✉❡ ❞✐ss❡❝t✐♦♥ ❜❧✉❡ tr✐❛♥❣✉❧❛t✐♦♥s ✷ ✶ ✶ ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❉ ◦ ✐s ❛ tr✐❛♥❣✉❧❛t✐♦♥ 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  12. ❆❝❝♦r❞✐♦♥ ❞✐ss❡❝t✐♦♥s ❉ ◦ ✐s ❛ tr✐❛♥❣✉❧❛t✐♦♥ = ⇒ ❜❧✉❡ ❞✐ss❡❝t✐♦♥ = ❜❧✉❡ tr✐❛♥❣✉❧❛t✐♦♥s � ✷ n ✶ � C n := n + ✶ n 1 ◦ 28 • 2 • 3 ◦ 27 ◦ 26 • 4 • 5 ◦ 25 ◦ 24 • 6 • 7 ◦ 23 ◦ 22 • 8 • 21 ◦ 9 ◦ 10 • 20 • 11 ◦ 19 ◦ 12 • 18 • 17 ◦ 13 ◦ 14 • 16 • 15 ◦ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  13. ❈❤❛♣♦t♦♥✱ ❙t♦❦❡s ♣♦s❡ts ❛♥❞ s❡r♣❡♥t ♥❡sts ✭✷✵✶✻✮ ❆r❡ ❙t♦❦❡s ♣♦s❡ts ❧❛tt✐❝❡s❄ ❨❊❙✿ ●❛r✈❡r ❛♥❞ ▼❝❈♦♥✈✐❧❧❡✱ ❖r✐❡♥t❡❞ ❋❧✐♣ ●r❛♣❤s ❛♥❞ ◆♦♥❝r♦ss✐♥❣ ❚r❡❡ P❛rt✐t✐♦♥s ✭✷✵✶✻✮ ❆r❡ ❙t♦❦❡s ❝♦♠♣❧❡①❡s r❡❛❧✐③❛❜❧❡ ❛s ♣♦❧②t♦♣❡s❄ ❨❊❙✿ ▼✳ ❛♥❞ P✐❧❛✉❞✱ ●❡♦♠❡tr✐❝ r❡❛❧✐③❛t✐♦♥s ♦❢ ❛❝❝♦r❞✐♦♥ ❝♦♠♣❧❡①❡s ✭✷✵✶✼ ✮ ❡❧❡♠❡♥ts ♦❢ ❙t♦❦❡s ♣♦s❡ts s❡r♣❡♥t ♥❡sts ❄ ❨❊❙✿ ▼✳✱ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡ ♦♥ ❛❝❝♦r❞✐♦♥ ❝♦♠♣❧❡①❡s ✭✷✵✶✼ ✮ ❍✐st♦r② ❇❛r②s❤♥✐❦♦✈✱ ❖♥ ❙t♦❦❡s s❡ts ✭✷✵✵✶✮ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

  14. ❨❊❙✿ ●❛r✈❡r ❛♥❞ ▼❝❈♦♥✈✐❧❧❡✱ ❖r✐❡♥t❡❞ ❋❧✐♣ ●r❛♣❤s ❛♥❞ ◆♦♥❝r♦ss✐♥❣ ❚r❡❡ P❛rt✐t✐♦♥s ✭✷✵✶✻✮ ❨❊❙✿ ▼✳ ❛♥❞ P✐❧❛✉❞✱ ●❡♦♠❡tr✐❝ r❡❛❧✐③❛t✐♦♥s ♦❢ ❛❝❝♦r❞✐♦♥ ❝♦♠♣❧❡①❡s ✭✷✵✶✼ ✮ ❨❊❙✿ ▼✳✱ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡ ♦♥ ❛❝❝♦r❞✐♦♥ ❝♦♠♣❧❡①❡s ✭✷✵✶✼ ✮ ❍✐st♦r② ❇❛r②s❤♥✐❦♦✈✱ ❖♥ ❙t♦❦❡s s❡ts ✭✷✵✵✶✮ ❈❤❛♣♦t♦♥✱ ❙t♦❦❡s ♣♦s❡ts ❛♥❞ s❡r♣❡♥t ♥❡sts ✭✷✵✶✻✮ ❆r❡ ❙t♦❦❡s ♣♦s❡ts ❧❛tt✐❝❡s❄ ❆r❡ ❙t♦❦❡s ❝♦♠♣❧❡①❡s r❡❛❧✐③❛❜❧❡ ❛s ♣♦❧②t♦♣❡s❄ #( ❡❧❡♠❡♥ts ♦❢ ❙t♦❦❡s ♣♦s❡ts ) = #( s❡r♣❡♥t ♥❡sts ) ❄ ❚❤✐❜❛✉❧t ▼❛♥♥❡✈✐❧❧❡ ✭▲■❳✱ P♦❧②t❡❝❤♥✐q✉❡✮ ❚❤❡ s❡r♣❡♥t ♥❡st ❝♦♥❥❡❝t✉r❡

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